A 21-year-old man has been arrested by police investigating a sustained barrage of threats and abuse directed against a feminist campaigner on Twitter after she successfully campaigned for a woman's picture to be put on a new banknote.

Caroline Criado-Perez has faced a deluge of hostile tweets, including threats to rape and kill her, prompting the shadow home secretary, Yvette Cooper, to write to Twitter to criticise its "inadequate" response to the "disgraceful, appalling and unacceptable" comments.

The Metropolitan police said officers acting on their behalf in Manchester had arrested a man on suspicion of harassment offences. The Met said the arrest was in connection with an allegation of malicious communications received by officers in Camden last Thursday.

Criado-Perez tweeted that she was at a police station making a formal statement and there were "many more threats" to report.

A campaign in Criado-Perez's support calling for Twitter to introduce a button to allow speedy reporting of abuse has received thousands of signatures, and she has received support from MPs and celebrities. In response Tony Wang, the general manager of Twitter UK, said the company took online abuse seriously and called on people to report any "violation of the Twitter rules".

Cooper wrote to Wang on Sunday saying the company's response was inadequate. "Despite the scale and seriousness of these threats, the official response from Twitter continues to be extremely weak – simply directing Caroline away from Twitter towards the police, and, belatedly, directing users to abuse-reporting forms on Twitter.

"Of course it is right to report such abuse to the police, and it is very important that they investigate and pursue this case. But social media platforms also have a responsibility for the platform they give users. And in particular they have a responsibility not to tolerate this kind of abuse, rape threats and potentially criminal behaviour."

She added: "The response by Twitter has clearly been inadequate and fails not only Caroline but many more women and girls who have faced similar abuse on your social network."

Cooper said more than 20,000 people had signed an online petition asking Twitter to allow users to report abuse directly with one click. "I urge you to go further and ensure that Twitter carries out a full review of all its policies on abusive behaviour, threats and crimes, including more help for Twitter users who experience abuse, a clear complaints process and clear action from Twitter to tackle this kind of persecution," she said.

Attempts are being made to organise a boycott of the social media platform on 4 August to highlight the issue.

Earlier Criado-Perez said: "It's sadly not unusual to get this kind of abuse but I've never seen it get as intense or aggressive as this. It's infuriating that the price you pay for standing up for women is 24 hours of rape threats. We are showing that by standing together we can make a real difference. We made the Bank of England change its mind; we can do the same with Twitter."

Criado-Perez, a freelance journalist who co-founded thewomensroom.org.uk and the Week Woman blog, and fellow campaigners were delighted last week when the Bank of England confirmed that Jane Austen would replace Charles Darwin on £10 notes, probably in 2017.

Criado-Perez organised a campaign which included a petition signed by more than 35,500 people after the Bank decided to replace Elizabeth Fry with Winston Churchill on new £5 notes. The move would have meant there were no women apart from the Queen on Bank of England banknotes.

Stella Creasy, the Labour MP for Walthamstow, said: "What Caroline has had to deal with in the past day is not only disgusting but criminal. A quick look at Twitter shows that women are not prepared to stand by and take this kind of abuse. Twitter needs to get its house in order, and fast."

Wang tweeted: "We encourage users to report an account for violation of the Twitter rules by using one of our report forms. Also, we're testing ways to simplify reporting, eg within a tweet by using the 'report tweet' button in our iPhone app and on mobile web. We will suspend accounts that, once reported to us, are found to be in breach of our rules."

A Twitter spokeswoman added: "The ability to report individual tweets for abuse is currently available on Twitter for iPhone and we plan to bring this functionality to other platforms, including Android and the web. We don't comment on individual accounts. However, we have rules which people agree to abide by when they sign up to Twitter. We will suspend accounts that, once reported to us, are found to be in breach of our rules."

• This article was amended on 31 July 2013 because the original said that without the Jane Austen banknote, there would have been no women apart from the Queen on sterling banknotes. Two women can be found on Clydesdale Bank notes, Mary Slessor and Elsie Maud Inglis.